Indeed, if \(\phi = c \mathrm {e}^{-\pi (\cdot )^2}\) , with \(c = 2^{1/4}\) , it is well-known that (see Lemma 1.5.2 on p. 18 of [1]})
\(\mathcal {V}_\phi \phi (x,\omega ) = \left( \phi , \operatorname{M}_\omega \operatorname{T}_x \phi \right) = \mathrm {e}^{-2\pi \mathrm {i}x \omega } \left( \phi , \operatorname{T}_x \operatorname{M}_\omega \phi \right) = \mathrm {e}^{-\pi \mathrm {i}x \omega } \mathrm {e}^{-\frac{\pi }{2}\left( x^2 + \omega ^2 \right)},\) 
